2.5 SPM Practice (Long Questions)

Question 8:
(a) Complete the table in the answer space for the equation y = x³ – 4x – 10 by writing down the values of y when x = –1 and x = 3.

(b) For this part of the question, use graph paper. You may use a flexible curve rule.
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 10 units on the y-axis, draw the graph of y = x³ – 4x – 10 for –3 ≤ x ≤ 4 and –25 ≤ y ≤ 38.

(c) From your graph, find
(i) the value of y when x = 2.2,
(ii) the value of x when y = 15.

(d) Draw a suitable straight line on your graph to find the values of x which satisfy the equation x³ – 12x – 5 = 0 for –3 ≤ x ≤ 4 and –25 ≤ y ≤ 38.

Answer:

x	–3	–2	–1	0	1	2	3	3.5	4
y	–25	–10		–10	–13	–10		18.9	38

Solution:
(a)
y = x³ – 4x – 10  
when x = –1,
y = (–1)³ – 4(–1) – 10
= –7

when x = 3,
y = (3)³ – 4(3) – 10
= 5

(b)




(c)
(i) From the graph, when x = 2.2, y = –8
(ii) From the graph, when y = 15, x = 3.4


(d)
y = x³ – 4x – 10 ----- (1)
0 = x³ – 12x – 5 ----- (2)
(1) – (2) : y = 8x – 5

The suitable straight line is y = 8x–5. Determine the x-coordinates of the two points of intersection of the curve y = x³ – 4x – 10 and the straight line y = 8x –5.

x	0	2
y = 8x – 5	–5	–11

From the graph, x = –0.45, 3.7.